How do you find the roots for #f(x) = 15x^12 + 41x^9 + 13x^2 –10# using the fundamental theorem of algebra?
1 Answer
The FTOA tells us it has
Explanation:
The fundamental theorem of algebra tells you that a polynomial of degree
A corollary of this, often stated as part of the FTOA is that a polynomial of degree
To see why this follows, note that if we have one zero
In our example:
#f(x) = 15x^12+41x^9+13x^2-10#
is of degree
That's all the FTOA tells you. It does not help you find the zeros.
What else can we find out about the zeros of this polynomial?
Note that the signs of the coefficients follow the pattern:
#+ + + -#
With one change of sign, that means that there will be exactly one positive Real zero.
Reversing the sign on the term of odd degree we get the pattern:
#+ - + -#
With three changes of sign, that means that there will be either
We can try to find rational zeros using the rational root theorem:
Any zero of
That means that the only possible rational zeros are:
#+-1/15# ,#+-2/15# ,#+-1/5# ,#+-1/3# ,#+-2/5# ,#+-2/3# #+-1# ,#+-5/3# ,#+-2# ,#+-5# ,#+-10#
None of these works, so
Ultimately, we are stuck with using numerical methods (e.g. Durand-Kerner) to find approximations for the zeros:
#x~~-1.38923#
#x~~0.735661#
#x~~-0.775742+-0.203247i#
#x~~-0.487897+-0.736584i#
#x~~0.14955+-0.928279i#
#x~~0.714459+-1.21625i#
#x~~0.726412+-0.459408i#